Solving Polynomial and Rational Inequalities Algebraically. Approximating Solutions to Inequalities Graphically

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1 10 Inequalities Concepts: Equivalent Inequalities Solving Polynomial and Rational Inequalities Algebraically Approximating Solutions to Inequalities Graphically (Section 4.6) 10.1 Equivalent Inequalities Definition 10.1 Two inequalities are equivalent if they have the same solution set. Operations that Produce Equivalent Inequalities. Add or Subtract the same value on both sides of the inequality. Multiply or Divide by the same positive value on both sides of the inequality. Multiply or Divide by the same negative value on both sides of the inequality AND change the direction of the inequality. Example 10.2 For each pair of inequalities, determine if the two inequalities are equivalent. x 2 +2x 5 and x 2 +2x x < 5 and x > 1 x x+2 > 3 and x x+2 3 > 0 x > 3 and x > 3(x+2) x+2 You should NEVER multiply both sides of an inequality by an expression involving x that you don t know the sign of. Why? 1

2 10.2 Solving a Linear Inequality Example 10.3 (Linear Inequality) Solve the inequality 5 2x > x + 3 algebraically. Write the solution set in interval notation. Note 10.4 There are many ways to solve this inequality algebraically. We will begin by using addition and subtraction to move all the nonzero quantities to one side. This is not necessary for this inequality, but it will help us to understand the process needed for solving more complicated inequalities. Let s try to understand this solution graphically. The picture on the left contains the graphs of y = 5 2x and y = x + 3. The INTERSECT option was used to find the point of intersection of these two graphs. The picture onthe right contains the graphof y = 5 2x (x+3). The ZERO option was used to find the x-intercept of the graph. How can you approximate the solutions of an inequality graphically? 2

3 10.3 Polynomial and Rational Inequalities Thinking graphically can help us understand the algebraic procedure for solving Polynomial and Rational Inequalities. Let f(x) be a function. f(x) is positive when the graph of f is. f(x)isnegativewhenthegraphoff is. Example 10.5 (Polynomial Inequality) The graph of y = (x+3) 2 (x 1)(x 5) is shown below. The viewing window is [ 10,10] [ 200, 100]. Use the graph to help you approximate the solutions of the inequality (x+3) 2 (x 1)(x 5) > 0. The algebraic procedure for solving a polynomial inequality is based on the intuition we gain from the graphical solution. Algebraic Procedure for Solving Polynomial Inequalities 1. Use addition and subtraction to move all nonzero quantities to one side. 2. Find the roots of the polynomial. (If you can factor the polynomial, this can help.) 3. Make a sign chart to determine if the values between the roots lead to positive or negative values of the polynomial. 4. Answer the question. Example 10.6 Use the algebraic approach to solve (x+3) 2 (x 1)(x 5) > 0. 3

4 The procedure for solving rational inequalities is very similar to the procedure for solving polynomial inequalities with one exception. Polynomial graphs can only switch from being below the x-axis to being above the x-axis or vice-versa at an x-intercept. Rational graphs can switch sides at an x-intercept or a vertical asymptote. Moreover, regardless of the type of the inequality, you cannot include any x-value that will produce a zero in a denominator. If an x-value is associated with a vertical asymptote, it must always have an open circle on the number line and, hence, a parenthesis in the interval notation if it involved in the solution. Algebraic Procedure for Solving a Rational Inequalities 1. Use addition and subtraction to move all nonzero quantities to one side. 2. Simplify the rational expression so that it is a single fractional expression. 3. Find the roots of the numerator of the rational expression. These are possible x- intercepts. (If you can factor the polynomial, this can help.) 4. Find the roots of the denominator of the rational expression. These are possible vertical asymptotes. (If you can factor the polynomial, this can help.) 5. Make a sign chart to determine if the values between the roots found in 3 and 4 lead to positive or negative values of the rational expression. 6. Answer the question. Make sure that you never include values found in 4 in your answer. Example 10. (Rational Inequality) Solve 2 x+3 4 x 1 4

5 Now let s look at the graph of y = 2x 14. Does this graph reinforce the solution (x+3)(x 1) you found to the previous example? Look at Examples 10 and 11 in Section 4.6 of your textbook for more examples of rational inequalities. 5

Solving Linear and Rational Inequalities Algebraically. Definition 22.1 Two inequalities are equivalent if they have the same solution set.

Inequalities Concepts: Equivalent Inequalities Solving Linear and Rational Inequalities Algebraically Approximating Solutions to Inequalities Graphically (Section 4.4).1 Equivalent Inequalities Definition.1